Multilayered Elastic Media Research Articles

A superposition scheme to obtain fundamental boundary element solutions in multi‐layered elastic media

A superposition scheme is proposed to obtain a fundamental solution for boundary elements in multi-layered elastic media. A three-layered elastic region is obtained by superposing two sets of bonded half-planes and subtracting one infinite plane. Therefore, the solution for an element in a layered media can be expressed in terms of bonded half-plane solutions and an infinite-plane solution. The major advantages of this superposition scheme are: (1) it is unnecessary to introduce elements at the interface, (2) it can be extended to higher-order element, and (3) it may be applicable to three dimensions easily. The accuracy and performance of the developed model is illustrated by two examples. For the problem of a pressurized two-dimensional crack within a three-layered system, the comparison with other numerical results shows the model is quite accurate and efficient. The model is also used for a study of a practical two-dimensional mining problem in South Africa, i.e. stoping through a dyke with material properties different from the host rock. Copyright © 2000 John Wiley & Sons, Ltd.

Windowedf-kspectra of a three-dimensional wavefield excited by a point source in a two-dimensional multilayered elastic medium

SUMMARY When full 3-D modelling is too costly or cumbersome, computations of 3-D elastic wave propagation in laterally heterogeneous, multilayered 2-D geological structures may enhance considerably our ability to predict strong ground motion for seismological and engineering purposes. Towards this goal, we extend the method based on the combination of the thin-layer finite-element and boundary-element methods (TLFE-BEM) and calculate windowed f -k spectra of the 3-D wavefield. The windowed f -k spectra are spatially localized spectra from which the local properties of the wavefield can be extracted. The TLFE-BEM is particularly suited for calculating the complete wavefield where surface waves are dominant in multilayered media. The computations are performed in the frequency domain, providing the f -k spectra directly. From the results for the 3-D wavefield excited by a point source in a 2-D multilayered, sloped structure, it can be said that the phase velocity of the fundamental-mode Rayleigh wave in a laterally heterogeneous multilayered medium, estimated from the windowed f -k spectra, varies with the location of the point source. For the model calculated in this article, the phase velocity varies between the value for the flat layered structure of the thick-layer side and that for the structure just under the centre of the window. The exact subsurface structure just under the centre of an array in a laterally heterogeneous medium cannot be obtained if we use the f -k spectral analysis assuming a flat layered structure.

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.

On generalized Kelvin solutions in a multilayered elastic medium

This paper presents fundamental singular solutions for the generalized Kelvin problems of a multilayered elastic medium of infinite extent subjected to concentrated body force vectors. Classical integral transforms and a backward transfer matrix method are utilized in the analytical formulation of solutions in both Cartesian and cylindrical coordinates. The solution in the transform domain has no functions of exponential growth and is invariant with respect to the applied forces. The convergence of the solutions in the physical domain is rigorously and analytically verified. The solutions satisfy all required constraints including the basic equations and the interfacial conditions as well as the boundary conditions. In particular, singular terms of the generalized Kelvin solutions associated with the point and ring types of concentrated body force vectors are obtained in exact closed-forms via an asymptotic analysis. Numerical results presented in the paper illustrate that numerical evaluation of the solutions can be easily achieved with very high accuracy and efficiency and that the layering material inhomogeneity has a significant effect on the elastic field.

A normal mode model for multilayered acoustoelastic ocean environments based on an analytic reflection coefficient method

A normal mode model has been developed for underwater acoustic propagation in an ocean environment with multilayered elastic media below and/or above the water column. The compressional (p-) and shear (s-) wave sound speeds may either be constant or have a gradient (1/c2 linear with depth) in each layer. Mode eigenvalues are found by analytically computing the downward- and upward-looking plane-wave reflection coefficients R1 and R2 at a reference depth in the water and searching the complex k plane for points where R1R2=1. The complex k-plane search is greatly simplified by following the path along which ‖R1R2‖=1. The ‖R1R2‖=1 path connects all the modes, which are found as points on the path where the phase of R1R2 is a multiple of 2π. The direction of the path is found from the analytic derivative d(R1R2)/dk. Leaky modes are found, and multiple channels are handled. The eigenvalue finding algorithm appears to be robust and efficient. [Work supported by the ARL Independent Research and Development Program.] a)On leave from Appl. Res. Lab., The Univ. of Texas at Austin. b)On leave from Dept. of Phys., Private Bag 92019, Univ. of Auckland, Auckland, New Zealand.

Forward ultrasonic scattering from multidimensional solid or fluid inclusions buried in multilayered elastic structures

The scattering of elastic waves by multidimensional objects buried in a multilayered elastic background medium insonified by ultrasonic forces is studied. The framework developed is based on an integral equation formalism for elastic scattering, where a Born approximation for an inhomogeneous background medium is applied to obtain a closed-form expression for the scattered field. Both compressional and shear waves are allowed to propagate inside the layers, and take into account cross-polarization at the boundaries and multiple reflections within the layers to compute the scattered field. The procedure has been tested in cases where the Born approximation is severely put to test (100% contrast in shear modulus).

Static response of an orthotropic stratified medium to surface loads—strip loadings

Abstract The static deformation of an orthotropic multilayered elastic medium as a result of strip-loadings is studied. The transfer matrix approach is used to find the displacements and stresses at any point of the medium. Both plane strain and antiplane strain cases are discussed in detail. Two kinds of problems are considered; first, in which the multilayered medium lies on an elastic half-space with which it is in welded contact, and, second, in which the multilayered medium lies on a rigid base. The procedure developed is straightforward, convenient for numerical computation and avoids the complicated nature of the problem.

Synthetic seismograms using multifold path integrals - II. Computations

Summary. Synthetic seismograms for a laterally variable multilayered elastic medium are computed by using multifold Kirchhoff-Helmholtz path integrals. Multifold path integrals (MFPI) are a generalization of elastic Kirchhoff-Helmholtz (KH) integrals and were devised to circumvent the failure of the latter at caustics of source/receiver ray-fields on the surface(s) of integration. MFPI uses geometrical optics, plane wave generalized reflection and transmission coefficients, and an elastodynamic form of the KH integral. Such integrals can be evaluated numerically either in the frequency domain, by using a multidimensional generalized Filon quadrature formula, or directly in the time domain. We discuss two time domain techniques and show some examples of 2-D synthetics obtained by our algorithm. We also compare some of our results with those obtained by the finite difference technique. Unlike many other asymptotic methods, the MFPI method computes the signals diffracted from corners on any of the surfaces of material discontinuity. Careful examination of the source/receiver ray-field often enables one to reduce the fold of an MFPI by replacing the integrals over surfaces that do not have corners with their stationary phase point values.

Synthetic seismograms using multifold path integrals - I. Theory

In a laterally varying multi-layered elastic medium the amplitudes of reflected waves often cannot be correctly modelled by geometrical acoustics because of the presence of caustics, shadows and diffractions. Maslov theory and the Gaussian beam method give correct amplitudes at caustics but do not give correct arrival times for waves diffracted from a corner on a reflector. The multi-fold path integral (MFPI) method presented in this paper is an alternative that addresses these problems. The MFPI method uses a combination of Kirchhoff-Helmholtz theory and geometrical acoustics to propagate waves between interfaces, and plane-wave reflection/transmission coefficients to propagate them across the interfaces. The generalized ray associated with an MFPI consists of a number of smooth segments (one segment across each layer) joined together at interfaces. The integrations are over the point at which this generalized ray touches each interface. We derive MFPI's for PP. SS and PS reflections and for a PP refraction. The resulting formulas are given in detail and from them the MFPIs for other seismic phases can easily be obtained by inspection. We also discuss how a given MFPI can be numerically evaluated in the frequency domain or in the time domain. For models in which velocities vary between interfaces, much time can be saved by the use of layer matrices that give travel times and spreading factors as functions of points on adjacent interfaces. Finally, the MFPI method is compared to other generalized ray theories.

Kirchhoff-Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium - II. Computations

Summary. In a medium consisting of elastic layers with irregular interfaces, Kirchhoff-Helmholtz (KH) theory can be extended to synthesize the motion due to various generalized rays. An exact elastic form of the KH integral is first derived, then various asymptotic approximations are used to convert this integral into one which can be rapidly evaluated to give the motion of a single generalized ray. The approximations used are those of geometrical optics, for propagation across layers, and the Kirchhoff or tangent-plane approximation for propagation across boundaries. It is shown how the KH method leads naturally to a generalization of our usual notion of elastic reflection and transmission coefficients. The new coefficients are functions of both angle of incidence and angle of reflection or transmission and they are derived so as to obtain coordinate-free formulae that show clearly their relation to the conventional Snell’s law coefficients. The elastic KH method is applied first to the problem of a single interface, where its performance is compared to that of the Gaussian beam and Maslov methods. (For synthesizing reflections from irregular interfaces the KH method is superior because it includes signals diffracted from corners. However, when the interface is very smooth on the scale of a wavelength the Maslov and Gaussian beam methods are superior because they do not break down when there is a caustic on the reflector.) KH theory is then applied to a multilayered elastic medium and it is shown how the effects of frequency-dependent attenuation and dispersion can be incorporated into the theory by taking advantage of the approximately logarithmic variation of slowness with frequency in most earth materials. The limitations of the KH theory are discussed and some recent attempts to overcome these difficulties are reviewed. A new method for overcoming the problem of a caustic on the reflector becomes apparent when the KH integral is regarded as a member of a larger family of equivalent 1-fold integrals all of which are derivable from the same multifold path integral. Refracted or diving rays can be treated within the same formalism with equal benefit. For velocity models that are independent of one spatial direction (strike) a method is given for approximately converting 2-D results into 3-D results.

Kirchhoff?Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium ? I. Theory

Summary. In a medium consisting of elastic layers with irregular interfaces, Kirchhoff-Helmholtz (KH) theory can be extended to synthesize the motion due to various generalized rays. An exact elastic form of the KH integral is first derived, then various asymptotic approximations are used to convert this integral into one which can be rapidly evaluated to give the motion of a single generalized ray. The approximations used are those of geometrical optics, for propagation across layers, and the Kirchhoff or tangent-plane approximation for propagation across boundaries. It is shown how the KH method leads naturally to a generalization of our usual notion of elastic reflection and transmission coefficients. The new coefficients are functions of both angle of incidence and angle of reflection or transmission and they are derived so as to obtain coordinate-free formulae that show clearly their relation to the conventional Snell’s law coefficients. The elastic KH method is applied first to the problem of a single interface, where its performance is compared to that of the Gaussian beam and Maslov methods. (For synthesizing reflections from irregular interfaces the KH method is superior because it includes signals diffracted from corners. However, when the interface is very smooth on the scale of a wavelength the Maslov and Gaussian beam methods are superior because they do not break down when there is a caustic on the reflector.) KH theory is then applied to a multilayered elastic medium and it is shown how the effects of frequency-dependent attenuation and dispersion can be incorporated into the theory by taking advantage of the approximately logarithmic variation of slowness with frequency in most earth materials. The limitations of the KH theory are discussed and some recent attempts to overcome these difficulties are reviewed. A new method for overcoming the problem of a caustic on the reflector becomes apparent when the KH integral is regarded as a member of a larger family of equivalent 1-fold integrals all of which are derivable from the same multifold path integral. Refracted or diving rays can be treated within the same formalism with equal benefit. For velocity models that are independent of one spatial direction (strike) a method is given for approximately converting 2-D results into 3-D results.

A numerical study ofP-,SV-, andSH-wave generation in a plane layered medium

abstractA solution for the surface displacements due to buried dislocation sources in a multi-layered elastic medium is found using the Haskell (1964) paper as a starting point and more importantly, for notation. Through the introduction of some simple matrix operations, the Haskell (1964) solution is made simultaneously more compact and computationally stable. Time histories are computed for a perfectly elastic medium by performing classical contour integration in the complex wavenumber plane. A new aspect in the evaluation of those contours is introduced because of the recognition of nonzero singularity contributions of the Hankel and modified Bessel functions at k = 0. Theoretical ground motion time histories are presented to show the usefulness of the formulation. The overall objective of this paper is to incorporate the modifications made since 1964 to the Haskell (1964) paper in an easily understandable, step-by-step development.

THEORETICAL REFLECTION SEISMOGRAMS FOR ELASTIC MEDIA*

AbstractTheoretical seismograms for an explosive source in a multilayered elastic medium are constructed by Fourier synthesis and plane wave superposition. The calculation scheme which builds up a reflection matrix layer by layer in the frequency and wave number domain allows the inclusion of attenuation and a choice of the level of internal multiples in each layer.Comparative calculations of theoretical seismograms for an elastic model and in the acoustic approximation, neglecting shear, show that the main differences arise at large offsets. The inclusion of shear waves leads to lower reflected P wave amplitudes at the end of the spread but only small amounts of converted phases.

Surface waves in multilayered elastic media. Part II. Higher mode spectra and spectral ratios from point sources in plane layered Earth models

Abstract Phase and amplitude spectra of Rayleigh and Love waves are presented for two Earth models, one oceanic and one continental shield. The spectra of the first three Rayleigh modes and the first four Love modes are tabulated for point sources at selected depths. These tables along with computer algorithms described here allow one to estimate the amplitude spectra at nontabulated source depths. The use of spectral ratios as a means of determining source depth is investigated. A source depth of 20 km is obtained for the Fallon earthquake of July 20 1962. This depth agrees with previous estimates but the technique requires a fault-plane orientation which differs from radiation pattern solutions.

Lumped mass method for Rayleigh waves

abstract A simple numerical method is developed for the analysis of generalized Rayleigh waves in multilayered elastic media. The method which completely avoids the use of displacement potentials leads to a simple eigenvalue problem which may be solved by generally available effective computer codes. Numerical results obtained by the method show excellent agreement with previously published solutions obtained by other theories.

Propagation of waves in a multilayered elastic medium and their dependence on source mechanisms

Abstract The stress equations of motion for a multilayered elastic medium subjected to body forces of general type have been solved in terms of three new variables which are functions of the displacement components and their space derivatives. The contributions of the different layers appear as separate 2 × 2 and 4 × 4 matrices whose elements are functions of the material constants and thickness of the particular layer. Some simple applications of the analysis have been discussed.

Point Source in a Layer Overlying a Semi-infinite Solid with Special Reference to Diffracted Waves

Summary The problem of a point source of dilatational waves in a homogeneous, isotropic, perfectly elastic layer overlying a homogeneous, isotropic, perfectly elastic halfspace has been discussed. The terms corresponding to various reflected waves are obtained from the representation of the point source as a summation of plane waves by making use of reflection coefficients. These are then integrated along the modified Sommerfeld contour and various minimum-time-path and diffracted waves are obtained. Numerical values of the elastic constants are inserted and the dependence of the amplitudes of these waves on frequency and sourcedepth observed. The amplitudes of the diffracted waves are compared with those of the minimum-time-path waves and it has been found that the contribution due to the former is always small as compared to that due to the latter. In fact the former are appreciable only when the frequency is small. Chopra (1957) discussed the disturbance due to a point source of dilatational waves situated inside a stratum bounded on either side by a halfspace. The application of the Bromwich method of expansion followed by evaluation along the Sommerfeld loops, resolved the disturbance into various reflected, refracted and diffracted waves. Chopra showed that the successive terms in the Bromwich expansion of displacement potentials in one of two halfspaces can be obtained from the representation of the point source as a summation of plane waves by multiplying the integrand in this representation with appropriate reflection and refraction coefficients and also changing the exponent suitably. This method of obtaining the successive terms in the Bromwich expansion of displacement potentials can be extended to problems involving line or point sources in multilayered elastic media. 2. Statement of the problem We have taken the problem of a point source of dilatational waves in a homogeneous, isotropic, perfectly elastic layer overlying a homogeneous, isotropic, perfectly elastic halfspace. The layer is assumed to be of uniform thickness. The waves starting from the point source as dilatational waves are multiply reflected . from the free surface and from the interface variously as dilatational and distortional

Surface waves in multilayered elastic media I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space

ABSTRACT A matrix formulation is used to derive integral expressions for the time transformed displacement fields produced by simple sources at any depth in a multilayered elastic isotropic solid half-space. The integrals are evaluated for their residue contribution to obtain surface wave displacements in the frequency domain. The solutions are then generalized to include the effect of a surface liquid layer. The theory includes the effect of layering and source depth for the following: (1) Rayleigh waves from an explosive source, (2) Rayleigh waves from a vertical point force, (3) Rayleigh and Love waves from a vertical strike slip fault model. The latter source also includes the effect of fault dimensions and rupture velocity. From these results we are able to show certain reciprocity relations for surface waves which had been previously proved for the total displacement field. The theory presented here lays the ground work for later papers in which theoretical seismograms are compared with observations in both the time and frequency domain.